Nuprl Lemma : fps-slice-slice
∀[X:Type]. ∀[r:CRng]. ∀[m,n:ℤ]. ∀[f:PowerSeries(X;r)].
([[f]_m]_n = if (n =z m) then [f]_m else 0 fi ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-slice: [f]_n,
fps-zero: 0,
power-series: PowerSeries(X;r),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
fps-slice: [f]_n,
fps-coeff: f[b],
subtype_rel: A ⊆r B,
nat: ℕ,
power-series: PowerSeries(X;r),
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
crng: CRng,
rng: Rng,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
fps-zero: 0
Lemmas referenced :
fps-ext,
fps-slice_wf,
ifthenelse_wf,
eq_int_wf,
power-series_wf,
fps-zero_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
bag-size_wf,
nat_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rng_zero_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
applyEquality,
because_Cache,
lambdaEquality,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
natural_numberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mforall{}[m,n:\mBbbZ{}]. \mforall{}[f:PowerSeries(X;r)].
([[f]\_m]\_n = if (n =\msubz{} m) then [f]\_m else 0 fi )
Date html generated:
2018_05_21-PM-09_55_55
Last ObjectModification:
2017_07_26-PM-06_32_49
Theory : power!series
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